Survival of a recessive allele in a Mendelian diploid model

TL;DR

This paper investigates the long-term survival of a recessive allele in a diploid population model, demonstrating that it can persist for significant periods under certain dominance and fitness conditions.

Contribution

It provides a rigorous proof that a recessive allele can survive for a substantial time in a Mendelian diploid model when the dominant allele is also the most fit.

Findings

Recessive alleles can survive for at least K^{1/4-a} time units.

Survival time depends on population size and dominance assumptions.

The model extends understanding of allele persistence in evolutionary genetics.

Abstract

In this paper we analyse the genetic evolution of a diploid hermaphroditic population, which is modelled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet et al., 2013 have shown that, on the mutation time-scale, the process converges to the Trait-Substitution Sequence of adaptive dynamics, stepping from one homozygotic state to another with higher fitness. We prove that, under the assumption that a dominant allele is also the fittest one, the recessive allele survives for a time of order at least K^{1/4-a}, where K is the size of the population and a>0.